Let $A_n$ be a subset of $\{1,2,\ldots,n\}$ obtained by retaining each integer independently with fixed probability $\theta\in(0,1)$, and let $L_n$ be the least common multiple of the integers in $A_n$. We prove a functional large deviation principle, a functional moderate deviation principle, and a Strassen-type functional law of the iterated logarithm for the process $(\log L_{\lfloor{nt}\rfloor})_{0\le t\le1}$. The large deviation rate function is given by an entropy contraction for geometric marks, while the moderate deviation rate function and LIL cluster set are described by the reproducing kernel Hilbert space associated with the Gaussian limit process.
We prove that for every fixed $\lambda>0$ and all sufficiently large $n$, any $z_1,\dots,z_n\in\C$ with $|z_j|\geq1$ satisfy $\max_{2\leq k\leq n+1}|\sum_j z_j^k|>e^{-\lambda n}$. Consequently, the $n$th root of the optimal maximum tends to $1$, so no constant $C>1$ in Erd\H{o}s 973 can exist. The proof combines a trun...
Let $f$ be a Steinhaus or Rademacher random multiplicative function. We use methods from the theory of critical chaos to improve on the best known upper bound for partial sums of random multiplicative functions. In particular, our results imply that for any $\varepsilon>0$, almost surely $$ \Big|\sum_{n\le x}f(n)\Big|...
Let $f$ be an extended Rademacher random multiplicative function (RMF). We show that, for every fixed deterministic function $V(x)$ tending to infinity, almost surely there are arbitrarily large $x$ for which \[ \sum_{n\leq x}f(n) \geq \frac{\sqrt{x}(\log\log x)^{1/4}}{V(x)}. \] The corresponding negative fluctuation h...
Let $\{Y_n; n\ge 1\}$ be a sequence of independent and identically distributed random variables with mean zero in Peng's framework of the sub-linear expectation space $(\Omega,\mathscr{H},\widehat{\mathbb E})$, and $S_n=\sum_{i=1}^nY_i$. In this paper, we establish a limit law of \begin{align*}\lim_{n\to \infty}\max_{k...
Li-Xin Zhang, Yongze Song· Scientia Sinica Mathematica· 0 citations
Let $(M,\tau)$ be a semifinite von Neumann algebra, let $J$ be a trace-preserving Jordan isomorphism, and let $(n_k)_{k\geq 1}$ be a random increasing sequence of integers obtained by selecting each integer $n\geq 1$ independently with probability $n^{-\alpha}$, where $0<\alpha<\frac12$. We show that, almost surely, fo...
Léonard Cadilhac, C. le Merdy, S. Zadeh· 0 citations
Let $M_n$ be an $n\times n$ matrix with independent uniform sign entries. We prove that there exist absolute constants $C,c>0$ such that, for all sufficiently large $n$, \[ \mathbb{P}\!\left( \left|\operatorname{Per}(M_n)\right| \ge e^{-Cn}\sqrt{n!} \right) \ge 1-n^{-c}. \] This confirms, up to the exponential scale, t...
Yiming Chen· 0 citations
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